{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R843",
  "content_hash": null,
  "slug": "z101-four-ap-free-claim-certified-interval-30-67",
  "type": "claim",
  "title": "The certified interval is 30 through 67",
  "summary": "An explicit 30-set supplies the lower endpoint. Exact incidence counting in the progression design gives the upper endpoint.",
  "relevance": "For Largest four-term-progression-free subset of Z_101, record z101-four-ap-free-claim-certified-interval-30-67 (“The certified interval is 30 through 67”) records a bound, answer, status fact, or structural consequence. The record states: An explicit 30-set supplies the lower endpoint.",
  "relevance_source": "recorded",
  "body": "Write \\(\\alpha(101,4)\\) for the requested maximum. The independently replayed bounds are\n\\[\n\\boxed{30\\leq\\alpha(101,4)\\leq67}.\n\\]\nThe lower bound is witnessed by\n\\[\n\\{0,10,18,23,27,29,35,37,39,40,45,47,48,49,56,61,65,68,69,70,72,76,78,79,84,85,87,91,93,95\\}.\n\\]\nThe executable artifact generates all 5,050 distinct modular four-term progressions and checks that none lies inside this set.\n\nFor the upper bound, let \\(H\\) be the four-uniform progression hypergraph. Every vertex lies in 200 edges, and every unordered pair lies in six edges. If \\(A\\) is independent and \\(j_E=|A\\cap E|\\), then \\(j_E\\leq3\\). Double-counting selected pairs and selected vertex-edge incidences gives\n\\[\n6\\binom{|A|}{2}=\\sum_E\\binom{j_E}{2}\\leq\\sum_Ej_E=200|A|.\n\\]\nFor positive \\(|A|\\), this yields \\(3(|A|-1)\\leq200\\), hence \\(|A|\\leq67\\). The exact value remains open within this record.",
  "status": "open",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "subsets of Z/101Z containing no four distinct terms x, x+d, x+2d, x+3d with d nonzero",
    "bounds": {
      "modulus": {
        "min": 101,
        "max": 101
      },
      "progression_length": {
        "min": 4,
        "max": 4
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.4171/EM/16",
      "locator": "Lorenz and Stephanie Halbeisen, Avoiding arithmetic progressions in cyclic groups, Sections 0 and 3; the order-101 incidence calculation is independently derived here"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.4171/EM/16",
    "locator": "Lorenz and Stephanie Halbeisen, Avoiding arithmetic progressions in cyclic groups, Sections 0 and 3; the order-101 incidence calculation is independently derived here"
  },
  "relations": [
    {
      "slug": "R841",
      "title": "Exact witness and incidence verifier",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R842",
      "title": "Literature audit and exact-search specification",
      "object_type": "attempt",
      "relation": "contextualizes",
      "direction": "incoming"
    },
    {
      "slug": "z101-four-ap-free",
      "title": "z101 four ap free",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}